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    Maximum Likelihood Estimation: ' (t, θ) = - X, θ), - X, θ) denotes the density of t X

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    Leonard Gonzalo Saavedra Astopilco
    This document discusses maximum likelihood estimation for duration models. It provides examples of hazard functions for the Weibull, lognormal, and exponential distributions. It then explains that the loglikelihood function for duration models consists of the sum of the log hazard and log survivor functions over all observations. Explanatory variables can be included in the model through an index function to relate the duration to covariates.

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    Maximum Likelihood Estimation: ' (t, θ) = - X, θ), - X, θ) denotes the density of t X

    Uploaded by

    Leonard Gonzalo Saavedra Astopilco
    12 views1 page
    This document discusses maximum likelihood estimation for duration models. It provides examples of hazard functions for the Weibull, lognormal, and exponential distributions. It then explains that the loglikelihood function for duration models consists of the sum of the log hazard and log survivor functions over all observations. Explanatory variables can be included in the model through an index function to relate the duration to covariates.

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    pg492

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    This document discusses maximum likelihood estimation for duration models. It provides examples of hazard functions for the Weibull, lognormal, and exponential distributions. It then explains that the loglikelihood function for duration models consists of the sum of the log hazard and log survivor functions over all observations. Explanatory variables can be included in the model through an index function to relate the duration to covariates.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    12 views1 page

    Maximum Likelihood Estimation: ' (t, θ) = - X, θ), - X, θ) denotes the density of t X

    Uploaded by

    Leonard Gonzalo Saavedra Astopilco
    This document discusses maximum likelihood estimation for duration models. It provides examples of hazard functions for the Weibull, lognormal, and exponential distributions. It then explains that the loglikelihood function for duration models consists of the sum of the log hazard and log survivor functions over all observations. Explanatory variables can be included in the model through an index function to relate the duration to covariates.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 1

    492

    Discrete and Limited Dependent Variables

    h(t)
    5.0
    ...
    .....
    ........
    ..
    ........
    ........
    .
    .
    4.5 .... Weibull, = 1, = 0.5
    .
    .
    .
    .
    .
    .......
    ...
    ........
    ...
    ........
    ........
    .
    .
    .
    .
    .
    .
    .
    .
    Weibull, = 1, = 1.5.............
    4.0 ...
    ...
    ...
    .......
    ..........
    ...
    ..
    .
    ......
    .
    .
    .......
    .
    .
    .
    .
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    .
    .
    .
    .
    .
    ......
    .
    .
    3.5 ...
    ...... ...................
    ...
    ...
    ................
    ...
    .....
    .
    ......
    1
    .....
    ....Lognormal,
    = 0, =
    3.0 ....
    ......
    .......
    .....
    ...
    3
    ...
    .
    ......
    .
    .
    .
    .
    .
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    .
    ........
    ...
    ...
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    ......
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    .........
    2.5 ...
    .....
    ..
    ...........
    .....
    ...
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    ......
    ....
    ...
    ............
    .....
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    ..............................
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    ...
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    2.0 ...
    .
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    .... ..........
    .
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    .
    .
    ............
    ...
    ..
    ........ .. ....
    ................ .............................. Weibull, = 1, = 1.25
    1.5 ....
    .
    ... ......... ............
    ... ................
    Exponential, = 1
    .
    .
    .
    .
    .
    1.0 ..........................................................................................................................................................................................................................................................................................................................................
    ............. .. ...............
    .
    ...........
    .............. Lognormal, = 0, = 1
    ........ ........
    0.5 .............. ... ........................ ........................................................................................................................
    ............................................................. ...........................................................................
    ..... ..
    .............................................
    ... ..
    .. ..
    .
    ....
    t
    0.0
    0
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    Figure 11.4 Various hazard functions

    Maximum Likelihood Estimation


    It is reasonably straightforward to estimate many duration models by maximum likelihood. In the simplest case, the data consist of n independent
    observations ti on observed durations, each with an associated regressor vector Xi . The loglikelihood function for t, the vector of observations with typical
    element ti , is just
    n
    X
    `(t, ) =
    log f (ti | Xi , ),
    (11.84)
    i=1

    where f (ti | Xi , ) denotes the density of ti conditional on the data vector


    Xi for the parameter vector . In many cases, it may be easier to write the
    loglikelihood function as
    `(t, ) =

    n
    X
    i=1

    log h(ti | Xi , ) +

    n
    X

    log S(ti | Xi , ),

    (11.85)

    i=1

    where h(ti | Xi , ) is the hazard function and S(ti | Xi , ) is the survivor function. The equivalence of (11.84) and (11.85) is ensured by (11.81), in which
    the hazard function was defined.
    As with other models we have looked at in this chapter, it is convenient to let
    the loglikelihood depend on explanatory variables through an index function.
    As an example, suppose that duration follows a Weibull distribution, with

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